Question

If matrix $A$ is given by $A=\left[\begin{array}{cc}6& 11\\ 2& 4\end{array}\right],$ then the determinant of $A^{2005}-6A^{2004}$ is

• A. $2^{2006}$
• B. $(-11)2^{2005}$
• C. ${7.2}^{2006}$
• D. $(-9)2^{2004}$

Answer 1

The correct option is B $(-11)2^{2005}$

$|A^{2005}-6A^{2004}|=|A{|}^{2004}|A-6I|$
$=2^{2004}\left|\begin{array}{cc}0& 11\\ 2& -2\end{array}\right|$
$=(-22)2^{2004}$
$=(-11)(2{)}^{2005}$